If the distance of points 2i^+3j^+λk^ from the plane r → ⋅( 3 i ^ +2 j ^ +6 k ^ )=13 is 5 units then λ=

MHT CET · Maths · Three Dimensional Geometry

If the distance of points $2 \hat{i} + 3 \hat{j} + λ \hat{k}$ from the plane $\vec{r} \cdot (3 \hat{i} + 2 \hat{j} + 6 \hat{k}) = 13$ is $5$ units then $λ =$

A $6, - \frac{17}{3}$
B $6, \frac{17}{3}$
C $- 6, - \frac{17}{3}$
D $- 6, \frac{17}{3}$

Verified Solution

Answer & Solution

Correct Answer

(A) $6, - \frac{17}{3}$

Step-by-step Solution

Detailed explanation

Equation of plane

\bar{r} \cdot (3 \hat{i} + 2 \hat{j} + 6 \hat{k}) = 13

i . e . 3 x + 2 y + 6z - 13 = 0

Given point

(2, 3, λ)

Distance of plane from the point

= |\frac{a x_{1} + b y_{1} + c z_{1} + d}{\sqrt{a^{2} + b^{2} + c^{2}}}|

\Rightarrow

5 = |\frac{3 (2) + 2 (3) + 6 λ - 13}{\sqrt{9 + 4 + 36}}|

\Rightarrow

5 = |\frac{6 λ - 1}{7}|

\Rightarrow 6 λ - 1 = \pm 35

\Rightarrow 6 λ = 36, 6 λ = - 34

λ = 6, λ = - \frac{17}{3}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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